I am always confused when I enter a place full of family members, and I greet Asalamu Alaikum to each member separately. Is this the correct way ?
Narrated Abu Huraira:
Allah's Messenger (ﷺ) said, "The riding person should greet the walking one, and the walking one should greet the sitting one, and the small number of persons should greet the large number of persons."
In his book, (al-Adhkaar),
Al-Nawawi (RA) said: Note that initiating the greeting of salaam is sunnah and mustahabb, but not obligatory. It is sunnah on the basis of kafaayah. If it is a group that is giving the greeting, it is sufficient for one of them to say the salaam, but if all of them say salaam, that is better. As for returning the greeting, if it is one person that is greeted, he is obliged, on an individual basis, to return the greeting, but if it is a group then responding to the greeting is a communal obligation (fard kafaayah) for them; if one of them responds, then there is no blame on the others, but if all of them fail to respond, then all of them have sinned. If they all return the greeting, this is best.
Thus, there is no compulsion in Islam for you to say salaam to each individual of the group.
Also refer this.